Theorem aev 1890

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1791. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 1968, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)

Assertion

Ref	Expression
aev	|- ( A. x x = y -> A. z w = v )

Distinct variable group:   x, y

Proof of Theorem aev

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem1 1886	. . 3 |- ( A. x x = y -> A. u u = w )
2		ax6ev 1721	. . . 4 |- E. u u = v
3		ax-7 1739	. . . . 5 |- ( u = w -> ( u = v -> w = v ) )
4	3	aleximi 1632	. . . 4 |- ( A. u u = w -> ( E. u u = v -> E. u w = v ) )
5	2, 4	mpi 17	. . 3 |- ( A. u u = w -> E. u w = v )
6		ax5e 1682	. . 3 |- ( E. u w = v -> w = v )
7	1, 5, 6	3syl 20	. 2 |- ( A. x x = y -> w = v )
8		axc16g 1887	. 2 |- ( A. x x = y -> ( w = v -> A. z w = v ) )
9	7, 8	mpd 15	1 |- ( A. x x = y -> A. z w = v )

Syntax hints:   -> wi 4   A.wal 1377   E.wex 1596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803

This theorem depends on definitions:  df-bi 185  df-ex 1597

This theorem is referenced by:  ax16nf  1891  axc16gALT  2079  2ax6e  2180  wl-naev  29575  wl-ax11-lem2  29619